Do law schools know when you've started their application in the LSAC system?

If so, does that affect if they offer you a merit (LSAT)-based fee waiver?

For example. I took the October LSAT. It's possible that some schools that I want to apply to may give me a fee waiver. If I start the application now, would the law school have less incentive to give a fee waiver?

I received fee waivers after starting applications for schools. I doubt schools can see whether or not you've started their applications, but even if they can, they still might find it advantageous to send you a fee waiver. I started far more applications than I submitted. A fee waiver could be the difference between merely starting an application and submitting it.